Kiro, Avner Taylor coefficients of smooth functions. (English) Zbl 1461.30009 J. Anal. Math. 142, No. 1, 193-269 (2020). Summary: We study the Borel map, which maps infinitely differentiable functions on an interval to the jets of their Taylor coefficients at a given point in the interval. Our main results include a complete description of the image of the Borel map for Beurling classes of smooth functions and a moment-type summation method which allows one to recover a function from its Taylor jet. A surprising feature of this description is an unexpected threshold at the logarithmic class. Another interesting finding is a “duality” between non-quasianalytic and quasianalytic classes, which reduces the description of the image of the Borel map for non-quasianalytic classes to the one for the corresponding quasianalytic classes, and complements classical results of Carleson and Ehrenpreis. Cited in 2 Documents MSC: 30B10 Power series (including lacunary series) in one complex variable 30D60 Quasi-analytic and other classes of functions of one complex variable Keywords:\(C^\infty\) functions; Taylor coefficients; Beurling classes; quasianalytic classes PDFBibTeX XMLCite \textit{A. Kiro}, J. Anal. Math. 142, No. 1, 193--269 (2020; Zbl 1461.30009) Full Text: DOI arXiv References: [1] Anderson, J. M.; Binmore, K. G., Closure theorems with applications to entire functions with gaps, Trans. Amer. Math. Soc., 161, 381-400 (1971) · Zbl 0224.30007 [2] Badalyan, G. V., Quasipower Series and Quasianalytic Classes of Functions (2002), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1026.26021 [3] T. Bang, Om quasi-analytiske Funktioner, PhD Thesis, University of Copenhagen, Copenhagen, 1946. [4] Bang, T., The theory of metric spaces applied to infinitely differentiable functions, Math. 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